I've long been confused by the two goats on Yom Kippur. The kohein gadol confesses his and his house's sins on the bull that is offered; in addition, there is a goat that is offered (without additional confession) and one that he confesses the people's sins on and then it's sent into the wilderness. In all other cases that I'm aware of, when we seek atonement we do it through an animal that is carefully slaughtered, not one that's sent out to an uncertain (but almost certainly messier) fate. And the torah could have had us confess over the goat that's offered (or include the people in the confession over the bull), but it didn't.

Is there a deeper meaning to the Azazel goat? It seems to introduce an element of uncertainty about our atonement -- you send this randomly-chosen goat out and who knows what happens? -- except that the mishna in Yoma describes binding it with a red cord (4:2) and later watching for (half of) that cord to turn white, so they do get an answer. Given that there are other atonement offerings too on Yom Kippur, how should we understand the Azazel goat? I am left wondering whether, like at the Pesach seder, we are instructed to do this in an unusual way in order to prompt questions and study. But all of this is speculation.

@Daniel off a cliff or into the wild; I've seen both interpretations. It's going to die, but it's not like slaughtering where we know it's happening right now and expediently. Maybe "uncertainty" wasn't the best word (though I said element of uncertainty); I'm describing a feeling, not just that the goat does end up dead.
– Monica Cellio♦Jun 5 at 14:53

4 Answers
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The goat [of the Day of Atonement] that was sent [into the wilderness] (Lev. xvi. 20, seq.) served as an atonement for all serious transgressions more than any other sin-offering of the congregation. As it thus seemed to carry off all sins, it was not accepted as an ordinary sacrifice to be slaughtered, burnt, or even brought near the Sanctuary; it was removed as far as possible, and sent forth into a waste, uncultivated, uninhabited land. There is no doubt that sins cannot be carried like a burden, and taken off the shoulder of one being to be laid on that of another being. But these ceremonies are of a symbolic character, and serve to impress men with a certain idea, and to induce them to repent; as if to say, we have freed ourselves of our previous deeds, have cast them behind our backs, and removed them from us as far as possible.

The ninth problem is presented by the sacrifice the Israelites were wont to offer up to Azazel on the day of Atonement. To certain people this name sounded like that of a demon. Our reply hereunto is that Azazel was the name of a mountain, and that names of this type were given to mountains is borne out by the statement of Scripture: And he took the rock by war, and called the name of it Joktheel (II Kings 14:7). Similarly Jabneel (Josh. 15:11), Irpeel (ibid. 18:27), and Jeruel (II Chron. 20:16) were all of them localities.

Now one of the two heads [of cattle] was offered up by the priests in the sanctuary in view of the fact that most of their sins were committed in the sanctuary. The other, again, was offered up by the nation outside of the sanctuary because most of their sins were committed outside of it, preference being thereby shown to the priests. As for the casting of the lots, which was, as it were, the most objectionable feature of the whole affair, let me explain that that was not due to any variation in the recipient of the sacrifice, for both were offered up to the same Master. Lots were cast only on account of the difference in the persons in whose behalf the sacrifices were offered; namely, that they were offered for priests and Israelites. It was, therefore, necessary to throw lots first, so that whatever fell to each one's share might be offered up in his behalf with the assurance that it belonged to him.

The Sefer HaChinuch explains (§95) that the two goats of Yom Kippur are symbolic for the choice presented to a sinner – the goat for Hashem representing one who repents, and the goat for Azazel representing one who doesn’t.

And about the sacrifice of Azazel that would be sent alive to a place of destruction and of extinction, we will say about the simple understanding of the matter [as follows]: The complete sinner should not imagine that after his soul receive the punishment for [the] sins, it will return to stand in the place of the good or [that] there will be some survival and good - even if he is like Yerovam the son of Nevat and his colleagues - just like he sees the whole entire year that the body of the animal that is an illustration for the body of the sinner has some remnants in the House of God, may He be blessed, in the ashes that stay there at the time of the burning. [As] they do not take them out from the Temple until after much time. Therefore in [this] living goat that carries all of the sins, they will see a hint that a sinner whose sins are great - like the heretics and those that deny Torah or the revival of the dead, and all of those that oppose Israel are included [as well] - will never see any good, and 'their worm will never die and their fire will never be extinguished.' [It is] like the procedure of this goat - [that] in his carrying the multitude of sins of all of Israel, is sent completely to a desolate land. He is not found in the House of God, not for slaughter and not for sprinkling - 'its memory will be lost from the earth.' And this is [the meaning of] what they, may their memory be blessed, said (Talmud Yerushalmi Yoma 6:3) that at the time that Israel was accepted, [the goat] would not reach halfway [down] the mountain before it would become [detached,] limb [from] limb - to show them the illustration of a compete sinner. As so will he be quickly destroyed, and he will be entirely, completely destroyed. [This is] in order that they would learn and [understand] the lesson and improve their ways. And this sign is good for them, as only he who loves one teaches him lessons, as it is written (Proverbs 13:24), "but he who loves him disciplines him early."

The Ramban begins by explaining that azazel refers to some sort of evil power (as opposed to the explanation of Rashi that it means a "hard and rocky place"). He first cites the Ibn Ezra, who hints that the explanation of azazel is found at the end of thirty-three. The Ramban, who, as we know, believes in not disclosing secrets, nevertheless here feels free to explain the Ibn Ezra, since, as he states, Chazal have already done so in several places. The reference of the Ibn Ezra is, as R. Chavel explains in a footnote, to the thirty-third verse following the present one, which is "and they shall no longer offer their sacrifices to the demons…" (17,7). The Ramban then cites Bereishit Rabba which associates the sa'ir hamishtalei'ach (the "sent goat") with Eisav ("ish sa'ir" – a hairy man). The Ramban, of course, understands this not as the historical figure of the brother of Yaacov, but as the twin and opposite figure to Yisrael, the power of evil. He makes this explicit in his next quote, from Pirkei d'Rabi Eliezer, that identifies the destination of the se'ir hamishtalei'ach as Samael. We now have only to understand who is Samael.

The quote from Pirkei D'Rabi Eliezer is as follows:

That is why on Yom Kippur they would give Samael a bribe to not cancel their sacrifice, as is written, "one lot to God and one lot to azazel," the lot of God is a burnt-offering, and the lot of azazel is a goat of sin-offering, and all the sins of Israel are on it. Samael sees that there is no sin in them on Yom Kippur. He says to God: Master of the worlds, you have one people on earth who are like the ministering angels in heaven – just as the ministering angels are barefoot, so Israel is barefoot on Yom Kippur; just as the ministering angels neither eat nor drink, so Israel does not eat or drink on Yom Kippur, just as the ministering angels cannot bend, so Israel stands all Yom Kippur; just as the ministering angels, peace serves as an intermediary between them, so Israel, peace serves as an intermediary between them on Yom Kippur; just as the ministering angels are free of all sin, so Israel is free of all sin on Yom Kippur.

God hears the testimony of Israel from their accuser and He atones for the altar, and for the Temple, and for the priests, and for all the congregation.

It is clear from this quote that azazel is Samael, who is the "accuser" of Israel; in other words, the satan. The Ramban goes on to explain.

See, they have told his name and his actions. And this is the secret of the matter. For they used to worship other gods, who are the angels, offering them sacrifices which are for them a sweet savor (rei'ach nicho'ach)…. Now the Torah totally prohibited the acceptance of their divinity or any worship of them, but God commanded that on Yom Kippur we send a goat to the desert to the prince who rules in desolate places, which is appropriate as he is the master of (that place), and from the emanation of his strength comes destruction and desolation, for he is the cause of the stars of the sword and blood and wars and quarrels and wounds and plagues and division and destruction, and, in general, the soul of the sphere of Mars. And his portion among the nations is Eisav, who is the nation who inherits the sword and wars. And among the animals, (his portion is) the goat, and in his portion are also the demons who are called mazikim in the language of the Rabbis, and se'irim in the language of Scripture, for both he and his nation are called se'ir (= goat, demon, and another name for Edom, the land of Eisav).

The intention of the se'ir hamishtalei'ach is not that it be an offering from us to him, God forbid, but rather that our intention should be that we are fulfilling the will of our creator who has commanded so. The parable for this is if one were to make a banquet for the master, and the master would command the one making the banquet: "Give a portion to this particular servant of mine." In this case, the one who makes the banquet is not giving anything to the servant, and is not honoring him, but rather everything is given to the master, and the master is giving a reward to the servant. And he has fulfilled his master's command and has done, in his master's honor, all that he was commanded. But the master, out of concern for the giver of the banquet, wished that all his servants take part in it, so that they praise him and not belittle him.

What does the service of Yom Kippur express? The Ramban gives a parable of a feast, to which the servant is invited so that he not be left out. I think that the meaning of this is that the service of Yom Kippur works by bringing about a unity of the different powers, the different servants, as it were, of God. On Yom Kippur, the Jews are like the angels, as the midrash in the Pirkei d'Rabi Eliezer makes clear. We make a banquet in God's honor, and it is important that all God's servants be included, because by partaking in the banquet they all become part of the unity of Israel's service of God. We don't serve the principle of evil, God forbid; we show that we understand that even the principle of evil is subject to God and dependent on Him. That is the opposite of serving evil; it is making, or rather demonstrating, that evil is subservient to God.

The bull was “devoured” by fire, and offered as a sacrifice to heaven, its fragrant flames rising towards the sky.

The goat was offered as a sacrifice to the merciless desert, which “devoured” it by other means, such as heat exhaustion, dehydration, starvation, animals of prey, etc.

The pagan ancestors of the Hebrews (I'll take it you are probably familiar with the pious oral tradition about Abraham destroying his father's idols, and then jokingly putting the hammer into the hands of the one he spared) most likely worshiped the actual physical elements of heaven and earth, but to latter monotheistic Jews this simply came to mean God's dominion over all creation (Deuteronomy 10:14).